I can't complete the companion's quest Purity. Vilkas asks me and I accept the mission. He follows me out of the building, then turns around and goes back in. In my journal, it shows the mission in the completed section but doesn't have it completed. It also won't let me pick up any followers and says I already have one.

I was hoping 1.5 would fix this. Didn't really read the patch notes till now. Any way around this?

Are you running any follower mods?

Have you tried fast traveling to where you are supposed to go and see if you shows up there?

Fixed Follower lite

I have tried fast traveling and it didn't work. I noticed that Farkas has an option for a job where he didn't before. Going to do that then come back and see what happens. I'll disable fixed follower and see if that works as well.

I had this happen to me once too and I'm on the ps3, so it's probably not mods. I think I somehow ended up accepting a radiant companions quest and the purity quest at the same time and no matter what I tried I couldn't unglitch it and had to reload.

I have exactly the same problem.

Do Farkas's quest first. Then Vilkas should follow.

He never showed up for me. I had the same glitch. I gave up. I did not, however, have any issues with followers. Apparently mine didn't glitch that badly.

JiSoo has it right: cure Farkas first then Vilkis. I'd tried various permutations and found I could cure myself without curing them but kept retrying etc and this is what worked for me.
Summary of getting it to trigger:
Do a radiant quest for Aela, Farkas and Vilkis. Have the Witches head/s in your inventory. Ask Farkas for a job, cure him. Repeat with Vilkis. Cure yourself.
Note: when you ask for a job it is when they are sat out back by the training dummies that they ask you to help cure them.

Question: does Aela ever ask for the cure?